Lipschitz regularity for scalar minimizers of autonomous simple integrals

[摘要] We prove Lipschitz regularity for a minimizer of the integral integral(a)(b) L(x,x')dt, defined on the class of the AC functions x: [a, b] --> R having x(a) A and x(b) = B. The Lagrangian L: R x R --> [0,+infinity] may have L(s,(.)) onconvex (except at xi = 0), while L((.),xi) may be non-lsc, measurability sufficing for xi not equal 0 provided, e.g., L**((.)) is lsc at (s,0) For Alls. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient psi/p((.)) (and not of L**((.)) itself, as usual), where psi(s)+p(s)h(xi) approximates the bipolar L**(s,xi) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-15

[关键词] calculus of variations;nonconvex nonlinear integrals;regularity properties